Econometrics-I-15 - Econometrics I Professor William Greene...

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Part 15: Generalized Regression Applications Econometrics I Professor William Greene Stern School of Business Department of Economics
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Part 15: Generalized Regression Applications Econometrics I Part 15 – Generalized                 Regression                 Applications ™  1/45
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Part 15: Generalized Regression Applications Leading Applications of the GR Model p Heteroscedasticity and Weighted Least Squares p Autocorrelation in Time Series Models p SUR Models for Production and Cost p VAR models in Macroeconomics and Finance ™  2/45
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Part 15: Generalized Regression Applications Two Step Estimation of the Generalized Regression Model Use the Aitken (Generalized Least Squares - GLS) estimator with an estimate of 1. is parameterized by a few estimable parameters. Examples, the heteroscedastic model 2. Use least squares residuals to estimate the variance functions 3. Use the estimated in GLS - Feasible GLS, or FGLS ™  3/45
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Part 15: Generalized Regression Applications General Result for Estimation When Is Estimated p True GLS uses [ X-1 X ] X-1 y which converges in probability to . p We seek a vector which converges to the same thing that this does. Call it “feasible” GLS, FGLS, based on ™  4/45 1 1 1 ˆ ˆ - - - X X X y Ω Ω
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Part 15: Generalized Regression Applications FGLS Feasible GLS is based on finding an estimator  which has the same properties as the true GLS. Example Var[e i]  =  s 2 [Exp( g ¢ zi )]2. True GLS would regress  y/[s  Exp( g ¢ zi )]  on the same  transformation of  xi . With a consistent estimator of [s , g ], say [s, c ], we  do the same computation with our estimates. So long as plim [s, c ] = [s , g ], FGLS is as “good”  as true GLS.        •   Consistent        •   Same Asymptotic Variance        •   Same Asymptotic Normal Distribution ™  5/45
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Part 15: Generalized Regression Applications FGLS vs. Full GLS VVIR (Theorem 9.6) To achieve full efficiency, we do not need an efficient estimate of the parameters in , only a consistent one. ™  6/45
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Part 15: Generalized Regression Applications Heteroscedasticity Setting: The regression disturbances have unequal variances, but are still not correlated with each other: Classical regression with hetero-(different) scedastic (variance) disturbances. yi = xi + i, E[i] = 0, Var[i] = 2 i, i > 0. The classical model arises if i = 1. A normalization: i i = n. Not a restriction, just a scaling that is absorbed into 2. A characterization of the heteroscedasticity: Well defined estimators and methods for testing hypotheses will be obtainable if the heteroscedasticity is “well behaved” in the sense that no single observation becomes dominant.
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